= Solution
The attractive potential has $\Delta>0$. Its time-independent <Schrödinger equation> is
$$
-\frac{\hbar^2}{2M}\chi''(x)-\frac{\hbar^2}{2M\Delta}\delta(x)\chi(x)=E\chi(x).
$$
Integrate from $-\varepsilon$ to $\varepsilon$. <Continuity> makes the delta term equal to $-\hbar^2\chi(0)/(2M\Delta)$, while the integral of $E\chi$ tends to zero as the interval shrinks. The second-derivative term integrates to the difference of first derivatives. Hence
$$
\boxed{\chi'(0^+)-\chi'(0^-)=-\frac{\chi(0)}{\Delta}}.
$$
This expresses the point interaction through a derivative jump while the <wavefunction> remains continuous.
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